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A kinetic model for the finite-time thermodynamics of small heat engines

We study a molecular engine constituted by a gas of $N \sim 10^2$ molecules enclosed between a massive piston and a thermostat. The force acting on the piston and the temperature of the thermostat are cyclically changed with a finite period $τ$. In the adiabatic limit $τ\to \infty$, even for finite size $N$, the average work and heats reproduce the thermodynamic values, recovering the Carnot result for the efficiency. The system exhibits a stall time $τ^*$ where net work is zero: for $τ<τ^*$ it consumes work instead of producing it, acting as a refrigerator or as a heat sink. At $τ>τ^*$ the efficiency at maximum power is close to the Curzorn-Ahlborn limit. The fluctuations of work and heat display approximatively a Gaussian behavior. Based upon kinetic theory, we develop a three-variables Langevin model where the piston's position and velocity are linearly coupled together with the internal energy of the gas. The model reproduces many of the system's features, such as the inversion of the work's sign, the efficiency at maximum power and the approximate shape of fluctuations. A further simplification in the model allows to compute analytically the average work, explaining its non-trivial dependence on $τ$.